Extend AC to E, make CE=BM, and connect DE (actually, rotate the triangle BDM to CDE).
In BDM and CDE triangle, it can be proved that
Angle MBD=DCE=90 degrees,
And BM=CE, BD=CD.
So the triangle BDM and CDE are congruent.
So DM=DE
In the triangle DMN and Deng
Proof angle MDN=EDN=60 degrees
And DM=DE, DN=DN.
So the triangle DMN and DEN are congruent.
So MN=NE=EC+CN=BM+CN.
2。 MN=CN-BM
Find point e on AC, make CE=BM, and then connect with DE.
Triangle BDM and CDE are congruent, triangle DMN and Deng Quan are congruent.
So MN=NE=CN-CE=CN-BM.